Motivated by applications in shared mobility, we address the problem of allocating a group of agents to a set of resources to maximize a cumulative welfare objective. We model the welfare obtainable from each resource as a monotone DR-submodular function which is a-priori unknown and can only be learned by observing the welfare of selected allocations. Moreover, these functions can depend on time-varying contextual information. We propose a distributed scheme to maximize the cumulative welfare by designing a repeated game among the agents, who learn to act via regret minimization. We propose two design choices for the game rewards based on upper confidence bounds built around the unknown welfare functions. We analyze them theoretically, bounding the gap between the cumulative welfare of the game and the highest cumulative welfare obtainable in hindsight. Finally, we evaluate our approach in a realistic case study of rebalancing a shared mobility system (i.e., positioning vehicles in strategic areas). From observed trip data, our algorithm gradually learns the users' demand pattern and improves the overall system operation.